the IC/2002/26 TIM abdus salam united nations educational, scientific and cultural international XA0202578 organization centre 5"- V- .- international atomic energy agency for theoretical physics BUOYANCY AND THERMOCAPILLARY DRIVEN CONVECTION FLOW OF ELECTRICALLY 'if CONDUCTING FLUID IN AN ENCLOSURE WITH HEAT GENERATION Md. Anwar Hossain and D.A.S. Rees i> 33/3 IC/2002/26 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS BUOYANCY AND THERMO CAPILLARY DRIVEN CONVECTION FLOW OF ELECTRICALLY CONDUCTING FLUID IN AN ENCLOSURE WITH HEAT GENERATION Md. Anwar Hossain* Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and D.A.S. Rees Department of Mechanical Engineering, University of Bath, BathBA2 7AY, United Kingdom. Abstract The effect of surface tension on unsteady laminar natural convection flow of a viscous incompressible fluid in a rectangular enclosure with internal heat generation and in presence of a uniform transverse magnetic field acting in the direction normal to the gravity has been investigated. The top horizontal surface of the rectangular cavity is assumed to be free and the bottom one insulated; whereas the left vertical wall is cold and the right one is uniformly hot. The equations are non-dimensionalized and solved numerically by an upwind finite difference method together with a successive over-relaxation (SOR) technique. The effects of heat generation together with the combined effects of the magnetic field and the surface tension are presented graphically in terms of isotherms, streamlines and velocity vector plots. The effects of varying the physical parameters on the rate of heat transfer from the heated surface of the enclosure are also depicted. The fluid here has Prandtl number Pr = 0.054 while the value of the Grashof number is 2xlO4. MIRAMARE - TRIESTE May 2002 f Senior Associate of the Abdus Salam ICTP. Author for correspondence. E-mail: [email protected] Nomenclature 1 Cp = Specific heat at constant pressure (J kg^K" ) G = gravitational acceleration (m/sec2) Gr = Rayleigh number W=enclosure height (m) Ha = Magnetic field parameter /fc=effective thermal conductivity of the media (W/mK) Ma = Maragoni number Nu = Nusselt number p = fluid pressure (Pa) Pr = Prandtl number t = time (s) T= temperature (°C) u = velocity in x-direction (m/s) V = velocity in ^-direction (m/s) x, y = Cartesian coordinates (H) X, Y = dimensionless coordinates P = Coefficient of thermal expansion of fluid (K'1) 6 = Dimensionless temperature A = Dimensionless heat absorption/generation parameter JU = Effective dynamic viscosity (Pa/s) v= Effective kinematic viscosity (/Mp) p = Fluid density at reference temperature (7*0) T= Dimensionless time \(f= Streamfunction (m2/s) Q = Dimensionless vorticity 1. Introduction The characteristics of heat and fluid flow for a configuration of isothermal vertical walls, maintained at different temperatures and with adiabatic horizontal walls, are well understood (Ostrach [1-2]). Less work has been carried out for more complex thermal boundary conditions, such as an imposed thermal gradient that is neither purely horizontal nor purely vertical. Shiralkar and Tien [3] investigated, numerically, natural convection in an enclosure with temperature gradients imposed in both the horizontal and vertical directions simultaneously. A stabilizing vertical temperature gradient was found to result in lower vertical velocities and the generation of secondary vortices at opposite corners. On the other hand, a destabilising vertical gradient leads to the destruction of stratification in the core and the formation of unstably stratified thermal layers adjacent to the upper and lower surfaces. Chao and Ozoe [4] investigated the problem of natural convection in an inclined box with half the bottom surface heated and half insulated, while the top surface was cooled. Anderson and Lauriat [5] analysed both experimentally and theoretically the natural convection due to one isothermal cold vertical wall and a hot bottom wall. Kimura and Bejan [6] studied numerically the convection flow in a rectangular enclosure with the entire lower surface cooled and one of the vertical walls heated. November and Nansteel [7] and Nicolas and Nansteel [8] performed experiments and numerical investigations on convection in a water filled enclosure with a single cold isothermal vertical wall and a partially heated bottom wall. Granzarolli and Milanez [9] computed the case of a heated bottom wall and isothermally cooled vertical walls. Recently, Velusamy et al. [10] investigated the steady two-dimensional natural convection flow in a rectangular enclosure with a linearly-varying surface temperature on the left vertical wall, cooled right vertical and top walls and a uniformly-heated bottom wall. In this latter investigation, mild natural convection was found to reduce the heat load to the cold walls, and for any value of aspect ratio it was also found that there exists a critical Rayleigh number for which heat load is a minimum. Recently Hossain and Wilson [11] investigated the problem posed in [10] for an enclosure filled with a fluid-saturated porous medium, along with generation of heat depending on the fluid temperature. Nowadays, considerable interest has been generated on interaction of buoyancy and surface tension driven convection; since, when a free liquid surface is present, variation in liquid surface tension at the free surface - due to temperature gradients- can induce motion within the fluid. This flow is also known as thermocapillary flows or the Maragoni convection flow. This combined convection flow driven by buoyancy and surface tension plays an important role in growth of crystals and material processing, especially in small-scale and low gravity hydrodynamics [12-14]. Combined buoyancy and thermocapillary convection flow or the Maragoni convection flow in a differentially heated cavity has been investigated numerically by Bergman and Ramadhyani [15] to show that surface tension significantly alters the buoyancy induced flow. Srinivasan and Basu [16] computed numerically the thermocapillary flow in a rectangular cavity during laser melting. The gas-liquid interface was assumed to be flat with a sinusoidal variation of temperature. Thereafter Basu and Srinivasan [17] numerically simulated a two-dimensional steady state laser melting problem in a cavity; whereas, Chen and Huang [18] conducted a similar study with a moving heat flux along the free surface. Carpentar and Homsy [19] studied the problem of combined buoyancy-thermocapillary convection flow in a square cavity with free surface heated differentially in the horizontal direction. Influence of thermocapillary forces on natural convection flow in a shallow cavity has also been investigated numerically by Hadid and Roux [20]. Recently, Rudraiah et al. [21] investigated the combined surface tension on buoyancy driven flow of a electrically conducting fluid in presence of a transverse magnetic field acting in the direction of gravity in a square cavity to see how this force damp the hydrodynamic movements; since, this is required to enhance crystal purity, increase compositional uniformity and reduce defect density. In the present investigation we revisit the problem posed in [21] for the flow of an electrically conducting fluid with heat generation filled in an enclosure in presence of uniform transverse magnetic field acting in the direction normal to the gravity field. Numerical simulations have been carried out for a fluid having of smaller value of Prandtl number (Pr = 0.054) at a Grashof number of Gr = 2xlO4. The Maragoni number, Ma, which depends on the thermocapillary force, ranges between 0.0 to 102, the Hartmann number, Ha, which depends on the transverse magnetic field, ranges from 0.0 to 50 while the heat generation parameter, X, ranges from 0.0 to 200.0. A detailed development of the present investigation is given in the subsequent sections. 2. Mathematical formulation Here we consider an unsteady two-dimensional natural convection flow of a viscous incompressible and electrically conducting fluid in a rectangular enclosure of height H as shown in Fig. 1. The right and the left walls are maintained at uniform temperatures TH and Tc respectively such that TH > Tc. The bottom and the top walls are considered to be adiabatic or insulated. We also bring into account the effect of a uniform volumetric heat generation, q"' [W/m3], in the flow region. We further assume that (1) The fluid is permeated with by a uniform magnetic field of strength Bo the direction of which is indicated in Figure 1. No external electric field is assumed to exist and the Hall effect of the magnetohydrodynamic is also considered. (2) The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected (Gavandet [1]) (3) The pressure work, Joule heating and viscous dissipation effects are negligible. (4) The surface tension, 0, varies linearly with temperature as given below: where To= (TH+ Tc)/2 is the mean of the temperatures of heated and cold surface, y [= (l/ob)(9o/37)] is the temperature coefficient of the surface tension, a0 is a reference surface tension and T is the temperature of the fluid in the cavity (5) the upper boundary is assumed to be flat and the fluid above the surface is assumed to be gas of negligible viscosity